I have the official PSU and it's connected to a brand new Raspberry Pi 3B+, I copied the official Raspbian on to an SD Card and inserted into Raspberry Pi 3B+, it ran fine on the first day, and I shut down using the drop down menu remotely using VNC, I didn't disconnect the power, now I wanted to turn it on, so I removed and inserted the plug at the outlet and it isn't showing anything on the monitor, the red light stays on, the green light sometimes flickers, sometimes it doesn't, even after reconnecting the power cord.

I ran fsck on the SD card using a card reader and everything came clean, so I'm at a loss about why this is not turning on.

I have:
Raspberry Pi 3B+ with official power supply
Sandisk Class 10 SD card with Raspbian Stretch full

There is a fault that can affect a pi3B+ were by the voltage regulator chip fails after the first boot of the pi.

If you have a meter you can test the pi to see if it has failed by testing between the first 2 pins on the gpio and ground one should read 5V and the other 3.3V , just be careful you don't short any of the pins or you could damage your pi if it has not failed.

If the 3.3V is missing then your pi has suffered this failure and you should contact your supplier.

We want information… information… information........................no information no helpThe use of crystal balls & mind reading are not supported

There is a fault that can affect a pi3B+ were by the voltage regulator chip fails after the first boot of the pi.

If you have a meter you can test the pi to see if it has failed by testing between the first 2 pins on the gpio and ground one should read 5V and the other 3.3V , just be careful you don't short any of the pins or you could damage your pi if it has not failed.

If the 3.3V is missing then your pi has suffered this failure and you should contact your supplier.

I don't have a meter, so I can't do what you suggested.

But thankfully, SD card seemed to have corrupted, after re-flashing it, it came up, but there were few errors at the boot up, disconnecting and reconnecting the power seemed to have sorted it out.

I have:
Raspberry Pi 3B+ with official power supply
Sandisk Class 10 SD card with Raspbian Stretch full

Same problem again, I left the Raspberry Pi on, just a short while ago I disconnected it's power cable at the Pi enclosure and now when I connected it, it is not showing up. Does the Pi have to be connected to a monitor, keyboard and mouse all the time?

I have:
Raspberry Pi 3B+ with official power supply
Sandisk Class 10 SD card with Raspbian Stretch full

Same problem again, I left the Raspberry Pi on, just a short while ago I disconnected it's power cable at the Pi enclosure and now when I connected it, it is not showing up. Does the Pi have to be connected to a monitor, keyboard and mouse all the time?

The problem is back again, this time I didn't even disconnect it from the power. It was on, now the Pi is not showing on HDMI, turning the outlet off for 10 seconds and then turning it on is also not getting it rebooted. The green light is not showing up. Normally it the green light would flicker for few seconds at bootup. Only stable red light is visible.

Why does this keep happening? This is irritating and seems like a waste of money.

I have:
Raspberry Pi 3B+ with official power supply
Sandisk Class 10 SD card with Raspbian Stretch full

I didn't even turn it off. It was "on" the whole time, I'm accessing it through VNC Viewer and it was not responding. When I connected it to monitor, it was showing blank. So as I was unable to access it, I though turning it off and on would solve it, but even then it wasn't showing up, the green light wasn't flickering. Just a red stable light.

I have:
Raspberry Pi 3B+ with official power supply
Sandisk Class 10 SD card with Raspbian Stretch full

Having turned it off previously without shutdown could mean that you corrupted the SD-card at that time, and this corruption can come and haunt you later.

Also, when not bought from an official distributor, but from a website, the SD-card can be, and quite often is, a fake, with much less capacity than is stated, in that case all bets are off about what can happen. The most popular brands are the ones most often faked.

I would start again with a new SD-card, from a reliable source, and make sure you do a legal shutdown each time, but especially after the first time, (as that time the card has had the most changes, and is the busiest) even wait a few seconds after shutdown to give the card time to really finish what it is doing before removing the power.